PSI - Issue 78

Salvatore Dario Di Trapani et al. / Procedia Structural Integrity 78 (2026) 2118–2125

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1. Initialization : The algorithm generates the structural base acceleration input, defines the response of the uncontrolled structure in terms of absolute acceleration variances ( 2 0, a x j  ), and initializes the full-sized system matrices (3 n × 3 n for M , C , and K , and 3 n × 1 for M ), with zeros for all STLCD-related terms. These matrices serve as the restore-point configuration. 2. 1 st iteration (placement of the 1 st STLCD) : The algorithm evaluates every combination of STLCD parameters (natural frequencies, damping ratios, head-loss coefficients) across all possible floors, temporarily reducing matrices by removing zero rows and columns corresponding to inactive DOF. The configuration that minimizes i Of (Equation 4) is identified, determining optimal parameters and installation floor. This optimal floor is excluded from subsequent iterations, and matrices and variances are updated accordingly, establishing a new baseline. 3. Subsequent iterations (placement of successive STLCD units): The procedure repeats iteratively, identifying the optimal parameters and placement of each successive STLCD unit, until one optimally tuned device is installed on every floor. 4. M-STLCD Control Strategy: Results of the Optimized Configuration To numerically evaluate the effectiveness and versatility of the M-STLCD strategy, the sequential optimization procedure has been applied to two benchmark structures, a three-story and a six-story building, under distinct base excitations. First, both buildings have been subjected to a real accelerogram recorded during the 1999 Kocaeli earthquake in Turkey. Subsequently, each building has been driven by a harmonic base acceleration in resonance with its fundamental mode. The three-story building model is characterized, from bottom to top, by lumped masses of 1 M = 6.0×10^4 kg, 2 M = 4.5×10^4 kg and 3 M = 3.0×10^4 kg (being 1 2 3 tot M M M M = + + the total building mass) and interstory stiffness values of 1 k = 4.0×10^7 N/m, 2 k = 2.1×10^7 N/m and 3 k = 2.0×10^6 N/m. The modal damping ratios are set to 5%, 2%, and 3% for the 1 st , 2 nd , and 3 rd modes, respectively, while the natural frequencies 1  = 7.50 rad/sec, 2  = 17.23 rad/sec, and 3  = 35.23 rad/sec indicate a relatively stiff system. The six-story building model adopts structural parameters previously used in the literature, and its fundamental frequencies also indicate a stiff system (Chen and Wu (2001)). For the M-STLCD strategy applied to the three-story building, each unit has been assigned a mass ratio of 4%, equally divided between the sliding container (2%) and the liquid (2%), resulting in a cumulative mass ratio of 12% relative to tot M . To achieve the same cumulative mass ratio of 12% for the six-story building, each M-STLCD unit has been assigned a mass ratio of 2%. The horizontal-to-total length ratio j  has been set to 0.85. Based on practical considerations, the optimal parameters of each unit were searched within the following ranges: , min max [0.64 ,1.36 ] c j     , with min  and max  being respectively the fundamental frequency and the highest natural structural frequency; , [0.005, 0.05] c j   , [1,80] j   and [3,50] j L  m. For comparison, the M-STLCD has been evaluated against optimized configurations of the Multi-Tuned Mass Damper (M-TMD) and the Multi-Tuned Liquid Column Damper (M-TLCD), with each unit assigned the same mass ratio as in the M-STLCD strategy, i.e., 4% for the three-story building and 2 % for the six-story building. Table 1 reports peak values of absolute accelerations and interstory drifts for the three-story building equipped with the final M-STLCD configuration, compared with analogous final configurations for the M-TMD and the M-TLCD. As can be observed, the M-STLCD strategy significantly reduces structural responses at all floors, decreasing accelerations at the 2 nd floor from 8.51 m/s² to 5.22 m/s² (39% reduction) and interstory drift between the 2 nd and 1 st floors from 0.0191 m to 0.0141 m (26% reduction), thus remaining competitive with the M-TMD. Peak values clearly indicate that the response reduction achievable with the M-TLCD is limited, slightly exceeding the uncontrolled structure drift at the 3 rd -to-2 nd interstory (0.0840 m). A similar trend can be observed in terms of absolute acceleration response reductions at the top floors of both the three-story and six-story buildings, where the M-STLCD and M-TMD consistently achieve meaningful response attenuation throughout the entire time history, whereas the reduction obtained by the M-TLCD remains less pronounced. Furthermore, under resonant excitation at the 1 st structural mode frequency, the M-STLCD again demonstrates